Jay proves to be trustworthy by presenting Lucille with a brand new dress when she tore hers at one of Gatsby's parties in chapter 3:
When I was here last I tore my gown on a chair, and he asked me my name and address - inside of a week I got a package from Croirier's with a new evening gown in it.
In chapter 4, Gatsby specifically recognizes the "lies" people report about him. He uses this line:
I'll tell you God's truth...
Gatsby's use of God's name suggests that he is indeed telling the truth.
After telling Nick an entire life's background, Gatsby pulls out a symbol of his history in the form of a medal of valor from Montenegro. Having this item to support his story gives it a reality that Nick might not have believed before.
Gatsby is hard to take. His "old sport" references, aversion for alcohol, acceptance for all, and believable stories all make him appear trustworthy, but the three instances listed above are most convincing: an actual object, envoking God's name, and acting with integrity more closely prove trustworthiness.